Ecology, 95(2), 2014, pp. 316–328
Ó 2014 by the Ecological Society of America
Patch definition in metapopulation analysis: a graph theory
approach to solve the mega-patch problem
KYLE C. CAVANAUGH,1,5 DAVID A. SIEGEL,2 PETER T. RAIMONDI,3
AND
FILIPE ALBERTO4
1
Smithsonian Environmental Research Center, Smithsonian Institution, Edgewater, Maryland 21037 USA
2
Earth Research Institute, University of California, Santa Barbara, California 93106 USA
Department of Ecology and Evolutionary Biology, University of California, Center for Ocean Health, Long Marine Lab,
100 Shaffer Road, Santa Cruz, California 95060 USA
4
Department of Biological Sciences, University of Wisconsin, Milwaukee, Wisconsin 53201 USA
3
Abstract. The manner in which patches are delineated in spatially realistic metapopulation models will influence the size, connectivity, and extinction and recolonization dynamics of
those patches. Most commonly used patch-definition methods focus on identifying discrete,
contiguous patches of habitat from a single temporal observation of species occurrence or
from a model of habitat suitability. However, these approaches are not suitable for many
metapopulation systems where entire patches may not be fully colonized at a given time. For
these metapopulation systems, a single large patch of habitat may actually support multiple,
interacting subpopulations. The interactions among these subpopulations will be ignored if the
patch is treated as a single unit, a situation we term the ‘‘mega-patch problem.’’ Mega-patches
are characterized by variable intra-patch synchrony, artificially low inter-patch connectivity,
and low extinction rates. One way to detect this problem is by using time series data to
calculate demographic synchrony within mega-patches. We present a framework for
identifying subpopulations in mega-patches using a combination of spatial autocorrelation
and graph theory analyses. We apply our approach to southern California giant kelp
(Macrocystis pyrifera) forests using a new, long-term (27 years), satellite-based data set of
giant kelp canopy biomass. We define metapopulation patches using our method as well as
several other commonly used patch delineation methodologies and examine the colonization
and extinction dynamics of the metapopulation under each approach. We find that the
relationships between patch characteristics such as area and connectivity and the demographic
processes of colonizations and extinctions vary among the different patch-definition methods.
Our spatial-analysis/graph-theoretic framework produces results that match theoretical
expectations better than the other methods. This approach can be used to identify
subpopulations in metapopulations where the distributions of organisms do not always
reflect the distribution of suitable habitat.
Key words: giant kelp; Landsat; metapopulation; modularity; network; patch dynamics; remote sensing;
spatial autocorrelation; synchrony.
INTRODUCTION
The mega-patch problem
The metapopulation approach provides a framework
to model the population dynamics of a set of interconnected subpopulations (Hanski 1999). As metapopulation theory has developed, scientists have moved from
spatially implicit models (Levins 1969, 1970) and simple
two-population models (Freedman and Waltman 1977,
Holt 1985) to spatially realistic multi-population models
(e.g., incidence function model; Hanski 1999). These
realistic metapopulation models are critical for predicting the dynamics of real populations (Hanski 1999). The
realistic metapopulation framework parameterizes patch
size, connectivity, extinction, and colonization, which
Manuscript received 2 February 2014; revised 3 June 2013;
accepted 15 July 2013. Corresponding Editor: M. H. Graham.
5 E-mail: [email protected]
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are all allowed to differ among patches. Metapopulation
theory predicts that extinction probability decreases as
the size of patches of habitat increase, because it
assumes that expected population size is positively
correlated with patch size (Hanski 1999). Larger
populations are expected to have a lower chance of
extinction due to demographic and environmental
stochasticity (Foley 1997). Theory also predicts that
the colonization rate of empty patches increases with
increasing connectivity, as inter-patch migration is
critical for recolonizing suitable, unoccupied habitat.
Increasing connectivity may also reduce the probability
of patch extinction through a mechanism called the
rescue effect (Brown and Kodric-Brown 1977, Hanski
1999).
Patch definition is pivotal to the construction of the
realistic metapopulation model, because it will determine patch size and connectivity, the latter being a
function of the size and distance of neighboring patches
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SOLVING THE MEGA-PATCH PROBLEM
(Hanski 1999). However, patch definition is often
arbitrary. Many empirical metapopulation studies select
some separation distance below which sites are grouped
into the same patch; however, the rationale for the
choice of separation distance is rarely given (e.g.,
Bradford et al. 2003, Reed et al. 2006). In some cases,
the definition of a patch is driven completely by the
structure of field sampling (McCauley et al. 1995,
Thomas and Kunin 1999). And in other cases continuous distribution of habitat makes patch definition
difficult or impossible (Thomas and Kunin 1999).
However, populations of organisms can be patchily
distributed even if the underlying habitat is not (Pannell
and Obbard 2003). For example, in metapopulations
with rapid population dynamics (e.g., algae, seagrasses,
oysters, mussels, annual terrestrial plants) it may be rare
for the species to colonize all available habitat during a
colonization and extinction cycle. As a result, large,
seemingly contiguous patches of habitat may support
multiple distinct and interacting subpopulations. The
interactions between these subpopulations will be
ignored if a ‘‘mega-patch’’ of habitat is defined based
on contiguous habitat or other commonly used patchdefinition methods. Hereafter we will refer to this
agglomeration of distinct subpopulations into a single
large patch as the ‘‘mega-patch problem.’’ We will refer
to distinct populations within a mega-patch as ‘‘subpopulations.’’ A goal of this paper is to develop a
method for breaking up these mega-patches into ‘‘subpatches,’’ smaller patches of habitat within the megapatch that support distinct subpopulations.
Current patch-definition algorithms exacerbate the
mega-patch problem. The most commonly used method
for delineating patches involves identifying contiguous
areas of habitat, hereafter called the ‘‘contiguity
approach.’’ Assuming that gridded habitat data are
available, contiguity can be restricted to grid cells that
touch in the four cardinal directions (four-cell rule) or
extended to cells that share a corner (eight-cell rule;
Turner et al. 2001). These rules are easy to implement,
however they are highly dependent on the resolution of
the data used to identify habitat (Girvetz and Greco
2007). For example, coarse maps with large cell sizes
may combine multiple patches with distinct population
dynamics. In addition, as already described, large,
contiguous patches of habitat can support multiple
distinct and interacting subpopulations, especially if the
metapopulation displays rapid turnover. A handful of
studies have extended the contiguity approach by adding
rules to account for variation in an organism’s
perception and behavioral characteristics (Theobald et
al. 2006, Girvetz and Greco 2007). These kinds of tools
generally merge patches separated by less than some
minimum distance that is selected based on the
organism’s movement or dispersal characteristics; hereafter we will refer to this approach as the ‘‘separationdistance’’ method. For instance, if the minimum distance
is set at 500 m, a distinct patch must be at least 500 m
317
from the nearest neighboring patch. When large areas of
patchy habitat contain many gaps that never quite
exceed the minimum distance, the resulting mega-patch
will cover a much larger area than the separation
distance and, therefore, defeat the original spatial
justification for a minimum distance.
There are three consequences for metapopulation
analysis when patch-definition methods result in megapatches: (1) reduction in mean population synchrony
within mega-patches, (2) reduction in the connectivity of
mega-patches, and (3) artificially low levels of mortality
for subpopulations within mega-patches. If multiple
distinct subpopulations are combined into a mega-patch
and treated as a single population, the synchrony in
population dynamics across the mega-patch (the correlation in abundance through time; Bjørnstad et al.
1999a) will be reduced. This reduction reflects signal that
is being lost. When the mega-patch is considered as a
single unit, it is impossible to analyze the mechanisms
driving variation in population dynamics among subpopulations within that mega-patch. Second, megapatches will have artificially low levels of connectivity.
When multiple subpopulations are combined into a
single mega-patch, there are fewer distinct patches in a
given neighborhood because they are absorbed into the
mega-patch. As a result, the mega-patch connectivity, a
function of the size and distance of neighboring patches,
can be severely reduced. Finally, patch mortality is
underestimated for mega-patches. Individual subpopulations within the mega-patch may go extinct and
recolonize, but because these subpopulations display
some level of asynchrony in their dynamics, the megapatch population persists for much longer than the
individual subpopulations.
The metapopulation approach is useful when ecological processes (dispersal and synchrony in population
dynamics) take place at two scales: the local, i.e., intrapatch, and metapopulation, i.e., inter-patch (Hanski
1999). Therefore, the goal of patch definition should be
to identify boundaries between these two spatial scales.
Patch edges should represent locations where (1) there
are barriers to dispersal (Jacobi et al. 2012) or (2) the
dynamics of a focal population differs from those of
surrounding populations. One way to identify distinct
subpopulations within mega-patches is by examining the
spatial scale of synchrony in a population (Bjørnstad et
al. 1999a). In order to account for the mega-patch
problem, we have developed a method to delineate local
patches with a combination of time series synchrony
analysis and network theory.
We use southern California giant kelp (Macrocystis
pyrifera) forests (see Plate 1) as a case study to examine
the impacts of patch delineation algorithms on the
extinction and colonization dynamics of metapopulations. Giant kelp represents an ideal model metapopulation system as kelp forests occur in discrete stands and
often exhibit spatially uncorrelated local dynamics
(Reed et al. 2006). In addition, kelp forest canopy
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KYLE C. CAVANAUGH ET AL.
FIG. 1. Map of study area in California, USA. The black
patches represent the composite kelp canopy (areas where kelp
appeared at least once during the study period, 1984–2011).
The study area consists of the entire map; the black box outlines
the region highlighted in Figs. 2, 4, and 6.
biomass distributions are available from remotely sensed
data covering the entire Southern California Bight at 30m spatial resolution every three months from 1984 to
2011 (following Cavanaugh et al. 2011). We demonstrate
that the way in which local populations are delineated
can impact whether empirical metapopulation studies
support theoretical predictions.
METHODS
Study system and empirical data
The metapopulation concept is particularly suitable
for populations of giant kelp (Macrocystis pyrifera).
Giant kelp individuals are sedentary after recruitment,
but coupled biophysical models (Reed et al. 2006) and
genetic variation among Macrocystis populations (Alberto et al. 2010, 2011) have indicated that ecologically
relevant amounts of spore exchange occur between
populations. Kelp is one of the few marine subtidal
species that has a floating canopy that can be observed
from above. As a result, remote-sensing tools can be
used to estimate the population size and temporal
dynamics of kelp populations across a range of spatial
scales (;10 m to .1000 km) scales (Cavanaugh et al.
2010, 2011).
Populations of giant kelp fluctuate greatly in space
and time and so high amounts of patch turnover can be
observed over relatively short periods of time. Short
lifespans of both kelp fronds (4–6 months) and entire
plants (2–3 years) combine with rapid growth (;2% of
total biomass per day) to produce a standing biomass
that turns over six to seven times per year (e.g., Reed et
al. 2008, 2011). Kelp forest dynamics are controlled by a
combination of environmental factors (e.g., light availability, wave disturbance, nutrient levels), herbivory,
and demographic processes (e.g., dispersal, recruitment,
intraspecific competition; reviewed in Foster and Schiel
Ecology, Vol. 95, No. 2
[1985] and Graham et al. [2007]). Local populations of
giant kelp go extinct and recolonize at irregular intervals
in response to these forcings. On large spatial scales,
climate cycles such as the El Niño–Southern Oscillation
can drive regional changes in kelp abundance. For
example, large waves and low nutrient conditions
associated with strong El Niño events in 1982–1983
and 1997–1998 caused widespread kelp extinctions
throughout most of southern California and Baja
California (Dayton and Tegner 1990, Edwards 2004,
Cavanaugh et al. 2011). The effects of smaller scale
disturbances such as more localized wave disturbance
and sea urchin grazing can also cause kelp populations
to display variability in rates of local extinction and
recolonization (Ebeling et al. 1985, Cavanaugh et al.
2013).
Local populations of giant kelp are connected
primarily by the dispersal of free-living microscopic
spore stages (Reed et al. 2006). Empirical studies and
physical transport modeling have shown that spore
dispersal in giant kelp populations routinely occurs on
scales from tens of meters to several kilometers (Reed et
al. 1988, Gaylord et al. 2002). These dispersal distances
are generally large enough to allow for connectivity
among local populations of giant kelp (Reed et al.
2006). Dislodged adult kelp plants (‘‘drifters’’) are
capable of producing viable spores after drifting many
kilometers (Macaya et al. 2005, Hernández-Carmona et
al. 2006); however, these spore sources do not appear to
play a major role in local population dynamics (Reed et
al. 2006).
We tracked giant kelp canopy biomass along the
mainland California coast from Pt. Sal to the United
States–Mexico border (approximately 550 km; Fig. 1)
from January 1984 to November 2011 using 30-m
resolution multispectral Landsat 5 Thematic Mapper
(TM) satellite imagery. Methods used to process and
calibrate the Landsat 5 TM imagery into kelp canopy
biomass density (kg/m2) are detailed in Cavanaugh et al.
(2011). We estimated canopy biomass over the entire
region from Landsat images taken approximately once
every 2–3 months. In order to place the time series of
kelp canopy biomass onto a regular time scale, we
calculated the mean canopy biomass for each season
from 1984–2011 (winter, January–March; spring, April–
June; summer, July–September; fall, October–December). We used the composite kelp canopy (area where
kelp canopy occurred at least once during the 27-year
time series; Figs. 1 and 2a) to approximate the
distribution of suitable habitat for giant kelp over our
study area. As a result, when we refer to a ‘‘patch’’ of
kelp in this paper, we are referring to habitat that kelp
has occupied at some point.
Modularity approach to patch definition
We utilized data on the spatial synchrony of kelp
populations and the spatial distribution of suitable kelp
habitat to hierarchically delineate subpopulations of
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319
FIG. 2. (a) Composite kelp canopy for the period 1984–2011 for a section of the coastline around San Diego and (b) example
snapshots of kelp canopy coverage during various seasons. The gray shaded patches in panel (b) represent the composite canopy,
while the black patches represent the habitat that was colonized during the season listed.
giant kelp within mega-patches. Our modularity-based
subpopulation identification methodology consists of a
series of ordered steps: (1) use the synchrony-distance
relationship of the organism to set a minimum
separation distance for mega-patches, (2) create a
network where nodes (cells where the species was
recorded at least once throughout the observation
period) are connected to other nodes by links that are
within the separation distance and have weights that are
defined by their synchrony, (3) divide the network into
sub-patches (network communities) by maximizing
modularity, a value that quantifies the goodness of fit
of a particular set of network subdivisions (Newman
2006), and (4) implement a sub-patch significance test to
merge patches with weak community structure.
In the first step, a minimum separation distance was
selected based on the spatial synchrony of the population. To determine the scale of synchrony in kelp
populations we modeled the relationship between
synchrony and distance for giant kelp canopy biomass
using the nonparametric correlation function (NCF,
Sncf function in R; Bjørnstad et al. 1999b). The NCF
uses a smoothing spline to estimate a continuous
function describing synchrony as a function of distance
(Bjørnstad et al. 1999a). We then modeled the kelp NCF
as an exponential decay function using least-squares fits
(Chiles and Delfiner 1999):
3x
ð1Þ
aþb 1e c
where x is the distance between pixels and a, b, and c are
the fit parameters. The parameter a represents the
modeled y-intercept value. The c parameter provides a
measure of the length scale of synchrony. The quantity a
þ b represents the synchrony value that the function
approaches as x (distance between patches in this case)
increases to infinity. Patches separated by less than this
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Ecology, Vol. 95, No. 2
FIG. 3. Conceptual diagram of subpopulations identified from a mega-patch network by the modularity optimization
procedure. In this example, the separation distance is set at 500 m. Habitat nodes (i.e., pixels) are linked to all other nodes that are
located within 500 m. Links between nodes are weighted based on the pairwise correlation in population fluctuations between
nodes. The thicknesses of the lines represent the weights of the links. Distinct subpopulations defined by the modularity
optimization process are identified by different shades of gray.
minimum distance were merged together. We performed
a sensitivity analysis to determine the impact of
changing this minimum separation-distance value.
Next, within each patch we created a network by
linking all habitat nodes within this minimum separation
distance (Fig. 3). The nodes were defined by the
minimum mapping unit of the habitat data, 30-m
Landsat pixels in our case study, but the technique can
be applied to any cell size. The links between nodes were
weighted based on the pairwise correlation in temporal
dynamics between nodes. We then divided the network
into sub-patches by optimizing the quality function
known as ‘‘modularity’’ over all possible divisions of the
network (Fig. 3). Modularity is defined as the number of
edges falling within groups minus the expected number
of within-group edges in a random network (Newman
2006). We used the Louvain method to efficiently
optimize the modularity in our network (modularity_
louvain_und function from the Brain Connectivity
Toolbox in Matlab; Zalesky et al. 2010). The Louvain
method is a greedy optimization method that can
rapidly detect community structure in very large
networks (Blondel et al. 2008).
Besides producing a vector containing each network
node (i.e., Landsat pixel) community assignment, the
optimization process produces a modularity value for
each network that describes how structured the original
mega-patch is (i.e., how well it breaks into sub-patches).
We filtered patches with weak community structure
(modularity ,0.1) by removing the sub-patch divisions
from these patches. We performed a sensitivity analysis
to examine the effect of varying this modularity
threshold. Finally, we used a simple test to further filter
patches with weak community structure. For each node
we calculated the number of that node’s intra sub-patch
links. For each sub-patch, we used a one-sided t test to
determine if there were more intra sub-patch links than
inter sub-patch links. If this was not the case then we
assigned each pixel of the sub-patch to the nearest
neighboring sub-patch and repeated the test.
We delineated patches from our giant kelp canopy
data set using our modularity method and two
commonly used patch-definition methods (contiguity
and separation distance). For each of the three patch
configurations we calculated the mean giant kelp canopy
biomass at each patch every season from 1984–2011
using the Landsat time series. A patch was considered
extinct when the total canopy biomass in the patch was 0
for .6 months. This 6-month threshold was used to
reduce the likelihood that a patch we considered
‘‘extinct’’ was actually populated by sub-surface juvenile
plants. Under most conditions, sub-surface juveniles will
reach the surface and form a canopy within 6 months
(Foster and Schiel 1985).
Statistical analysis
We calculated the number of patches and average
patch size for each of the patch delineation approaches.
We also calculated a number of metapopulation metrics
for each patch configuration: mean fraction of occupied
patches, mean patch persistence time, mean patch
extinction time, mean probability of extinction per
patch, and the mean probability of colonization per
patch.
We analyzed the connectivity of each patch in each
season of the study period. Connectivity (S ) was
calculated at each patch as
Si;t ¼
X Bj;t
j6¼i
dij2
ð2Þ
where dij was the straight-line (Euclidean) distance
between the closest points on patches i and j and Bj,t
was the biomass of patch j at season t (Moilanen and
Nieminen 2002, Reed et al. 2006). We examined the
relationships between patch habitat area and connectivity (Si,t) and the probabilities of patch extinction and
colonization using general linear models (GLMs) with
binomial error distributions and logistic link functions
(logistic regression). For the logistic regression predicting colonization, each patch that was extinct during a
given season represented a potential colonization event
(an observation in the logistic model). Patch habitat area
and connectivity were used to predict the probability of
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SOLVING THE MEGA-PATCH PROBLEM
321
FIG. 4. Distribution of patches delineated by three different patch-definition methods for a section of the San Diego coastline
from Encinitas to Pt. Loma. Distinct patches are identified by different colors, but the specific colors were chosen arbitrarily. In the
contiguity method, cells of habitat that shared a common border were grouped into patches. In the separation-distance approach,
patches separated by less than 500 m were merged. The modularity approach identified subpopulations by maximizing the
modularity of a network created using data on giant kelp spatial synchrony (see Methods).
colonization (1) or continued extinction (0) during the
following season. The logistic model predicting extinction was created the same way, except each colonized
patch in a given season represented a potential
extinction event. This model attempted to predict the
probability of extinction (1) or continued persistence (0)
during the following season. The independent variables,
patch area and connectivity, were tested for significance
based on likelihood ratio tests assuming that the
variables were chi-square distributed. We quantified
the relative importance of each independent variable as
the percentage increase in residual deviance when that
variable was removed from the full multiple regression
model.
RESULTS
Comparison of patch-definition approaches
Neither the contiguity nor the minimum-separationdistance approach adequately characterized patches of
habitat for giant kelp forests. In some parts of giant
kelp’s range, an almost continuous narrow band of
suitable habitat exists along the coastline (Figs. 1 and
2a). Both the contiguity and the minimum-separationdistance approaches described these long bands as single
large patches (e.g., large patches in southern part of Fig.
4a and 4b). However, the turnover of giant kelp
populations is rapid enough that forests rarely fully
colonized these large patches and, at any given time, the
realized distribution of kelp forests was a large number
of smaller sub-patches (Fig. 2b). This dynamic was
illustrated by the pattern of spatial synchrony in kelp
populations. The exponential function provided an
excellent fit to the kelp biomass NCF (r 2 ¼ 0.97, Fig.
5). Synchrony in giant kelp canopy biomass changes
decreased with increasing distance out to 500 m (c ¼ 520
m, Fig. 5). Therefore we selected 500 m as the minimum
separation distance for giant kelp populations. Many of
the patches created by both the contiguity and the
minimum-separation-distance approaches were substantially longer than 500 m (Fig. 4). Colonized sub-patches
within mega-patches of habitat were often separated
from each other by .500 m even if the distribution of
suitable habitat (determined by the composite kelp
canopy) was not (Fig. 2).
The number of patches delineated by the three patchdefinition methods ranged widely from 782 to 88 (Table
1). Mean patch size was inversely related to the number
of patches; the contiguity approach created the largest
number of patches and had the smallest mean patch size
while the separation-distance method resulted in the
least number of patches and the largest mean patch size
(Table 1). The contiguous canopy method created a
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KYLE C. CAVANAUGH ET AL.
FIG. 5. (a) Relationship between synchrony and distance
for giant kelp canopy biomass. Synchrony is defined as the
mean pairwise correlation between giant kelp populations over
the entire Landsat time series (1984–2011). The shaded region
represents the 95% confidence intervals. The solid line gives the
spatial nonparametric correlation function, and the dashed line
gives the modeled exponential fit. (b) The relationship between
rate of change in synchrony and distance for giant kelp canopy
biomass.
large number of small patches because individual pixels
and small groups of pixels separated from other patches
by small distances were treated as distinct patches. As a
result, mean patch size was low for this approach.
However, it also created mega-patches in areas with
large patches of contiguous habitat (Fig. 4a). This
resulted in a highly skewed distribution with a large
Ecology, Vol. 95, No. 2
number of very small patches (minimum patch size 9 3
104 km2) and a few very large patches (maximum patch
size 13 km2, Table 1). The 500-m separation-distance
method merged patches separated by ,500 m and so
created more mega-patches that spanned large sections
of the coastline. The largest mega-patch under this
approach was again 13 km2. The modularity optimization patch-definition method resulted in a distribution of
patch sizes that was less skewed than either the
contiguity or 500-m separation-distance methods (maximum patch size 2.5 km2, Table 1). This method created
249 patches with a mean patch size of 0.28 km2. The
modularity approach was relatively insensitive to small
changes in the maximum linkage distance and the
modularity threshold used to eliminate patches with
weak community structure (Appendix A: Fig. A1,
Appendix B: Fig. B1).
Fig. 6 illustrates how connectivity is underestimated
for mega-patches with classic patch-definition approaches. The two mega-patches in the southern portion of Fig.
6 have low connectivity values using the contiguous
habitat and separation-distance methods because there
are no large patches anywhere near each of the megapatches. If target patch area were incorporated into the
connectivity definition (Moilanen and Nieminen 2002),
then the connectivity of the mega-patches in Fig. 6a and
6b would be increased. However, patch area and patch
connectivity would then be strongly autocorrelated, and
it would be difficult to isolate the independent contribution of each of these variables. In contrast, in Fig. 6c
the mega-patches are broken up by the modularity
approach and each sub-patch has high connectivity
because other patches surround it (Fig. 6c).
Colonization probabilities varied between patch
configurations, ranging from 0.18 per quarter (contiguous canopy approach) to 0.44 per quarter (modularity
approach; Table 1). Extinction probabilities ranged
from 0.26 per quarter (modularity approach) to 0.49
per quarter (contiguous canopy approach). On average,
extinctions lasted for 2.5 and 5.2 years with the
modularity and contiguous canopy methods, respectively. Patches remained occupied on average for 1.7 and 4.2
years with the contiguous canopy and modularity
optimization methods, respectively (Table 1). The
contiguous canopy approach resulted in a large number
TABLE 1. Summary metrics for the colonization/extinction dynamics under each of the three patch-definition methods.
Metric
Contiguity
Separation distance 500 m
Modularity optimization
Number of patches
Mean patch area (km2)
Mean patch area, skewness
Mean patch alongshore length (km)
Mean nearest neighbor distance (km)
Mean occupancy fraction
Mean colonization probability (fraction per quarter)
Mean extinction probability (fraction per quarter)
Mean extinction length (yr)
Mean persistence length (yr)
782
0.089
16.22
0.18
0.11
0.25
0.18
0.49
5.2
1.7
88
0.792
4.15
1.26
1.21
0.46
0.3
0.33
3.5
4.5
249
0.28
3.02
0.74
0.26
0.58
0.44
0.26
2.5
4.2
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323
FIG. 6. Distribution of mean connectivity values for patches delineated by three different patch-definition methods for a section
of the San Diego coastline from Encinitas to Pt. Loma. For patch delineations, refer to Fig. 4.
of very small patches that displayed high extinction and
low colonization rates and a few mega-patches that
essentially never went extinct (e.g., the mega-patches in
the southern portion of Fig. 4a). The separationdistance approach created even more of these persistent
mega-patches. By separating these mega-patches into
sub-patches, the modularity approach better accounted
for the asynchrony in dynamics of subpopulations
within the same mega-patch.
The relationship between patch connectivity and the
probability of recolonization varied among the different
patch-definition methods. Connectivity had a significant
positive relationship with colonization under the contiguity and modularity methods, and a marginally
significant relationship with colonization under the
500-m separation-distance approach (Fig. 7; Appendix
C: Figs. C1, C2, and C3 give plots of the actual logistic
relationships between colonization/extinction and area/
connectivity). Patch area was clearly the most important
variable in explaining colonization probability under the
contiguous canopy and separation-distance approaches,
but connectivity was almost as important as patch area
under the modularity approach. Connectivity was
significantly negatively correlated with extinction probability under the modularity method, but was signifi-
cantly positively correlated with extinction probability
under the contiguity and 500-m separation-distance
methods (Fig. 7).
DISCUSSION
Patches are often defined arbitrarily in metapopulation studies. However, patch definition is a critical part
of metapopulation analysis and it can impact how well
empirical results reflect theoretical expectations. The
modularity optimization algorithm resulted in a more
even size distribution of patches as pixels separated by
small distances were combined into single patches and
large mega-patches were separated into multiple subpatches that better represented the spatial structure of
kelp canopy at a given snapshot in time. The average
alongshore length of patches created by the modularity
approach was 740 m (Table 1), which corresponded to
the length scales of local giant kelp population
synchrony (500 m, Fig. 5; Cavanaugh et al. 2013) and
the average distance of giant kelp spore dispersal
(meters to kilometers; Reed et al. 2006b, Alberto et al.
2010). The dispersal of giant kelp likely influences the
spatial scale of population synchrony (Cavanaugh et al.
2013). The modularity approach then incorporates
population synchrony into its patch-definition process
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Ecology, Vol. 95, No. 2
FIG. 7. Results of the generalized linear models for the probability of patch colonization and extinction under three different
patch-definition scenarios. The reduction in residual deviance and associated P value is given for each variable. White bars
represent positive effects, and shaded bars represent negative effects.
in order to create a more ecologically realistic patch
distribution. Large, contiguous mega-patches of habitat that appeared to act as multiple distinct sub-patches
(Fig. 2b) were indeed broken into sub-patches by the
modularity method (Fig. 4c). In contrast, both the
contiguous canopy and 500-m separation-distance
approaches created mega-patches that displayed within-patch variability in spatial synchrony, lower than
expected connectivity, and extremely high persistence.
Within-patch synchrony was variable in mega-patches
because their sizes were much larger than the scale of
synchrony in kelp populations. Also, there was no way
to account for intra-patch connectivity in these megapatches and so their connectivity levels were low. These
mega-patches were essentially a collection of nonsynchronous, interacting subpopulations, and so the
mega-patches almost never went extinct, even though
the individual subpopulations were highly dynamic.
The modularity approach had the highest occupancy
rates, highest colonization probabilities, and lowest
extinction probabilities. These characteristics were likely
a result of the modularity approach’s tendency to merge
small patches that were very close to one another and
break up large mega-patches. This would reduce the
number of very small ephemeral patches and increase
the number of highly connected patches. Although
extinction rates under the modularity approach were
lower than the other two approaches, they were still high
compared to some other models and observations. For
example, Burgman and Gerard (1990) developed a
stage-structured stochastic population model for giant
kelp and predicted an 80% chance that a local
population would go extinct over a 20-year time period.
This translates to a quarterly extinction probability of
just 0.01. Our extinction and recolonization probabilities
corresponded much more closely to empirical observations made by Reed et al. (2006). They observed
monthly extinction and recolonization rates of 0.06
and 0.08 (these correspond to quarterly extinction and
recolonization rates of 0.24 and 0.32). Our mean
occupancy fraction under the modularity approach
(0.58) also corresponded closely with the occupancy
fraction observed in the Reed et al. (2006) study
(;0.65).
We found that the choice of patch-definition
methodology can significantly impact the empirical
relationships between patch size and connectivity and
extinction and recolonization probabilities. Patch area
was significantly positively correlated with colonization
probabilities and negatively correlated with extinction
probabilities in all of the patch configurations (Fig. 7).
This result matched theoretical expectations: large kelp
forests may have a lower chance of stochastic
extinctions due to their large population size (Hanski
1999). In addition, the greater amount of suitable
habitat in large patches presents a larger target for
spores dispersing from nearby patches and so may
increase the probability that some part of the patch is
colonized. While the relationship between patch size
and extinction and recolonization probability was
consistent under the different patch-definition scenarios, connectivity explained more of the variability in
February 2014
SOLVING THE MEGA-PATCH PROBLEM
325
PLATE 1. (Top) Underwater view of a giant kelp (Macrocystis pyrifera) forest in southern California, USA. (Bottom) Floating
giant kelp canopy along the coast of central California, USA. Photo credits: F. Alberto.
extinction and colonization probabilities under the
modularity approach than it did under any of the other
approaches. These modularity results agreed with the
theoretical expectation that patch connectivity should
be positively correlated with colonization probability
(Hanski 1999). This indicates that outside spore
sources play a role in the colonization of empty patches
and help maintain the kelp metapopulation. There was
also evidence of a rescue effect under the modularity
approach, as highly connected patches demonstrated
lower extinction rates (Brown and Kodric-Brown
1977). However, under the separation-distance and
contiguity approaches, there was a positive relationship
between connectivity and extinction probability, in
contrast with theoretical expectations (Fig. 7). We
hypothesize that this is due to the creation of megapatches by these methods. The mega-patches had very
low extinction probabilities, but were not considered
highly connected because they absorbed the entire
nearby habitat.
While the total amount of variability in extinction and
colonization rates that was explained by patch area and
connectivity was low for all three scenarios, these
magnitudes are typical for empirical metapopulation
analyses (e.g., Franken and Hik 2004, Snäll et al. 2005,
Jönsson et al. 2008). There are likely a large number of
326
KYLE C. CAVANAUGH ET AL.
other variables that could cause variability in extinction/
colonization (e.g., wave exposure, depth, seafloor
rugosity and slope, nutrient availability, competition,
grazer abundance). In addition, there is some evidence
that kelp spores are capable of exhibiting arrested
development and may act as a ‘‘seed bank’’ for future
colonizations (Carney and Edwards 2010). These seed
banks could introduce noise into the connectivity–
recolonization relationship, as a patch’s recolonization
potential could be decoupled from its current connectivity level. Still, even with all of these potential
complicating factors, patch size and connectivity do
appear to influence extinction and colonization rates.
Empirical studies of the colonization and extinction
rates of sessile species are rare, but our results under the
modularity approach agree with other work that has
demonstrated the importance of connectivity in patch
dynamics (e.g., Verheyen et al. 2004, Snäll et al. 2005,
Jönsson et al. 2008). These results provide empirical
support for the importance of dispersal limitation, in
accordance with metapopulation theory (Hanski 1999),
even in marine systems where dispersal potential is often
much larger than it is in terrestrial systems (Kinlan and
Gaines 2003).
Network analytic techniques have been used to detect
community structure in social networks (Scott 2000),
metabolic networks (Jeong et al. 2000), the World Wide
Web (Kleinberg and Lawrence 2001), disease outbreaks
(Pastor-Satorras and Vespignani 2001), food webs
(Dunne et al. 2002), genetic data sets (Dyer and Nason
2004), marine metapopulations (Watson et al. 2011),
and many other systems. Here we have extended these
techniques to optimally characterize physical habitat for
metapopulation analysis (Bascompte 2007). This approach is an improvement over commonly used patchdefinition methods because it incorporates speciesspecific autocorrelation information that is a critical
part of the metapopulation framework (Hanski 1999).
As a result, our approach does not create mega-patches
every time habitat is contiguous or nearly contiguous
(Fig. 4c). This characteristic will be particularly important for realistically identifying patches in metapopulations with rapid population dynamics (e.g., algae, sessile
invertebrates, annual plants, etc.). In these systems, it
may be rare for the species to colonize the entire patch
during a colonization and extinction cycle. As a result,
large, seemingly contiguous patches may act as multiple
distinct and interacting sub-patches.
Recently, Jacobi et al. (2012) applied modularity to
identify subpopulations of a metapopulation, however
they grouped subpopulations based on connectivity
estimates rather than spatial autocorrelation patterns.
The present method does not require dispersal information to define the patches and the resulting patch
configurations can be independently compared to
information about the dispersal capabilities of the
species to determine if a metapopulation approach is
reasonable. This may have the added benefit of helping
Ecology, Vol. 95, No. 2
prevent the overuse of the metapopulation concept
(i.e., the case where a population has patchy habitat
but synchronous dynamics across its entire range,
Hanski and Gilpin 1997, Freckleton and Watkinson
2002). In our case, the mean inter-patch nearest
neighbor distance under the modularity approach was
280 m (Table 1). Empirical estimates of giant kelp
spore dispersal are on the order of meters to kilometers
(Reed et al. 2006b, Alberto et al. 2010), indicating that
the metapopulation approach is suitable for giant kelp.
There is also great potential to integrate the type of
dispersal data that Jacobi et al. (2012) used to identify
subpopulations with our data on local population
dynamics to improve models of demographic exchange.
Our approach incorporates synchrony in population
dynamics when identifying subpopulations. Measuring
population synchrony typically requires time series data
on population abundances. Historically, the availability
of spatially extensive time series data was limited, which
is one reason that many earlier meta-population studies
attempted to parameterize realistic metapopulation
models from a single snapshot of patch occupancy
(e.g., Hanski et al. 1996). Fortunately, time series data
on population dynamics is becoming increasingly
available through techniques such as long-term observational networks and retrospective remote sensing
surveys (e.g., Cavanaugh et al. 2011). In addition, this
approach does not necessarily require data that is both
spatially and temporally comprehensive; population
synchrony estimates made from spatially incomplete
data (e.g., point data from field surveys) could be
combined with static habitat maps (e.g., from historical
surveys or single-date aerial/satellite photos) to delineate
sub-patches using our method. In addition, this method
could be applied to other pairwise measures of
synchrony when demographic data is not available.
For example, patterns of population synchrony often
match patterns of synchrony in environmental variables
that drive population dynamics (Bjørnstad 2000).
Therefore, synchrony in environmental variables could
potentially be used as a proxy for demographic
synchrony. Spatial autocorrelation in connectivity and
dispersal patterns could also be used to partition megapatches (Jacobi et al. 2012).
The application of realistic metapopulation models is
dependent on the definition of patches. However, little
attention has been given to spatially explicit methods
for hierarchically defining patches based on the
dynamics of the focal species. Interestingly, many
studies skip over the problem of correctly identifying
patches that behave according to metapopulation
theory and instead concentrate their efforts on finding
appropriate connectivity metrics that explain the
extinction and colonization dynamics (e.g., Prugh
2009, Duggan et al. 2011, Magrach et al. 2012; an
exception is Jacobi et al. 2012). Here we have presented
a framework for delineating patches at multiple spatial
scales based on the spatial autocorrelation of the
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SOLVING THE MEGA-PATCH PROBLEM
population densities of the species in question. Future
theoretical and modeling studies are needed to further
refine our understanding of the ways that patch
definition can affect the results of metapopulation
analysis and modeling. Improving the methods for
delineating patches will help realistic metapopulation
models more accurately portray the systems they
are representing.
ACKNOWLEDGMENTS
We thank D. Reed and B. Kendall for helpful discussions
regarding this work. Financial support for this research was
provided by NASA’s Biodiversity and Ecological Forecasting
Science program, the National Science Foundation’s (NSF)
support of the Santa Barbara Coastal Long Term Ecological
Research (SBC LTER) project, and the by the NSF under the
Biological Oceanography grant (Award number: OCE1233839).
LITERATURE CITED
Alberto, F., P. T. Raimondi, D. C. Reed, N. C. Coelho, R.
Leblois, A. Whitmer, and E. A. Serrao. 2010. Habitat
continuity and geographic distance predict population
genetic differentiation in giant kelp. Ecology 91:49–56.
Alberto, F., P. T. Raimondi, D. C. Reed, J. R. Watson, D. A.
Siegel, S. Mitarai, N. C. Coelho, and E. A. Serrao. 2011.
Isolation by oceanographic distance explains genetic structure for Macrocystis pyrifera in the Santa Barbara Channel.
Molecular Ecology 20:2543–2554.
Bascompte, J. 2007. Networks in ecology. Basic and Applied
Ecology 8:485–490.
Bjørnstad, O. N. 2000. Cycles and synchrony: two historical
‘‘experiments’’ and one experience. Journal of Animal
Ecology 69:869–873.
Bjørnstad, O. N., R. A. Ims, and X. Lambin. 1999a. Spatial
population dynamics: analyzing patterns and processes of
population synchrony. Trends in Ecology and Evolution 14:
427–432.
Bjørnstad, O. N., N. C. Stenseth, and T. Saitoh. 1999b.
Synchrony and scaling in dynamics of voles and mice in
northern Japan. Ecology 80:622–637.
Blondel, V. D., J. L. Guillaume, R. Lambiotte, and E. Lefebvre.
2008. Fast unfolding of communities in large networks.
Journal of Statistical Mechanics 2008:P10008.
Bradford, D., A. Neale, M. Nash, D. Sada, and J. Jaeger. 2003.
Habitat patch occupancy by toads (Bufo punctatus) in a
naturally fragmented desert landscape. Ecology 84:1012–
1023.
Brown, J. H., and A. Kodric-Brown. 1977. Turnover rates in
insular biogeography: effect of immigration on extinction.
Ecology 58:445–449.
Burgman, M. A., and V. A. Gerard. 1990. A stage-structured,
stochastic population model for the giant kelp Macrocystis
pyrifera. Marine Biology 105:15–23.
Carney, L. T., and M. S. Edwards. 2010. Role of nutrient
fluctuations and delayed development in gametophyte
reproduction by Macrocystis pyrifera (Phaeophyceae) in
southern California. Journal of Phycology 46:987–996.
Cavanaugh, K. C., B. E. Kendall, D. A. Siegel, D. C. Reed, F.
Alberto, and J. Assis. 2013. Synchrony in dynamics of giant
kelp forests in driven by both local recruitment and regional
environmental controls. Ecology 94:499–509.
Cavanaugh, K. C., D. A. Siegel, B. P. Kinlan, and D. C. Reed.
2010. Scaling giant kelp field measurements to regional scales
using satellite observations. Marine Ecology Progress Series
403:13–27.
Cavanaugh, K. C., D. A. Siegel, D. C. Reed, and P. E.
Dennison. 2011. Environmental controls of giant-kelp
327
biomass in the Santa Barbara Channel, California. Marine
Ecology Progress Series 429:1–17.
Chiles, J. P., and P. Delfiner. 1999. Geostatistics, modelling
spatial uncertainty. Wiley-Interscience, New York, New
York, USA.
Dayton, P., and M. Tegner. 1990. Bottoms beneath troubled
waters: benthic impacts of the 1982–1984 El Niño in the
temperate zone. Elsevier Oceanography Series 52:433–472.
Duggan, J. M., R. L. Schooley, and E. J. Heske. 2011.
Modeling occupancy dynamics of a rare species, Franklin’s
ground squirrel, with limited data: are simple connectivity
metrics adequate? Landscape Ecology 26:1477–1490.
Dunne, J. A., R. J. Williams, and N. D. Martinez. 2002. Foodweb structure and network theory: The role of connectance
and size. Proceedings of the National Academy of Sciences
USA 99:12917–12922.
Dyer, J. D., and J. D. Nason. 2004. Population graphs: the
graph theoretic shape of genetic structure. Molecular
Ecology 13:1713–1727.
Ebeling, A. W., D. R. Laur, and R. J. Rowley. 1985. Severe
storm disturbance and reversal of community structure in a
Southern California kelp forest. Marine Biology 84:287–294.
Edwards, M. S. 2004. Estimating scale-dependency in disturbance impacts: El Ninos and giant kelp forests in the
northeast Pacific. Oecologia 138:436–447.
Foley, P. 1997. Extinction models for local populations. Pages
215–246 in I. A. Hanski and M. E. Gilpin, editors.
Metapopulation biology. Academic Press, San Diego,
California, USA.
Foster, M., and D. Schiel. 1985. The ecology of giant kelp
forests in California: a community profile. Biological report
85. United States Fish and Wildlife Service, Washington,
D.C., USA.
Franken, R. J., and D. D. Hik. 2004. Influence of habitat
quality, patch size, and connectivity on colonization and
extinction dynamics of collared pikas Ochotona collaris.
Journal of Animal Ecology 73:889–896.
Freckleton, R., and A. Watkinson. 2002. Large-scale spatial
dynamics of plants: metapopulations, regional ensembles and
patchy populations. Journal of Ecology 90:419–434.
Freedman, H. I., and P. Waltman. 1977. Mathematical models
of population interactions with dispersal. I: Stability of two
habitats with and without a predator. SIAM Journal on
Applied Mathematics 32:631–648.
Gaylord, B., D. C. Reed, P. T. Raimondi, L. Washburn, and
S. R. McLean. 2002. A physically based model of macroalgal
spore dispersal in the wave and current-dominated nearshore.
Ecology 83:1239–1251.
Girvetz, E. H., and S. E. Greco. 2007. How to define a patch: a
spatial model for hierarchically delineating organism-specific
habitat patches. Landscape Ecology 22:1131–1142.
Graham, M. H., J. A. Vasquez, and A. H. Buschmann. 2007.
Global ecology of the giant kelp Macrocystis: from ecotypes
to ecosystems. Oceanography and Marine Biology: an
Annual Review 45:39–88.
Hanski, I. 1999. Metapopulation ecology. Oxford University
Press, New York, New York, USA.
Hanski, I. A., and M. E. Gilpin. 1997. Metapopulation biology:
ecology, genetics, and evolution. Academic Press, San Diego,
California, USA.
Hanski, I., A. Moilanen, T. Pakkala, and M. Kuussaari. 1996.
The quantitative incidence function model and persistence of
an endangered butterfly metapopulation. Conservation Biology 10:578–590.
Hernández-Carmona, G., B. Hughes, and M. H. Graham.
2006. Reproductive longevity of drifting kelp Macrocystis
pyrifera (Phaeophyceae) in Monterey Bay, USA. Journal of
Phycology 42:1199–1207.
Holt, R. D. 1985. Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat
distribution. Theoretical Population Biology 28:181–208.
328
KYLE C. CAVANAUGH ET AL.
Jacobi, M. N., C. André, K. Doos, and P. R. Jonsson. 2012.
Identification of subpopulations from connectivity matrices.
Ecography 35:1004–1016.
Jeong, H., B. Tombor. R. Albert, Z. N. Oltvai, and A. L.
Barabási. 2000. The large-scale organization of metabolic
networks. Nature 407:651–654.
Jönsson, M. T., M. Edman, and B. G. Jonsson. 2008.
Colonization and extinction patterns of wood-decaying fungi
in a boreal old-growth Picea abies forest. Journal of Ecology
96:1065–1075.
Kinlan, B., and S. Gaines. 2003. Propagule dispersal in marine
and terrestrial environments: a community perspective.
Ecology 84:2007–2020.
Kleinberg, J., and S. Lawrence. 2001. Network analysis: the
structure of the web. Science 294:1849–1850.
Levins, R. 1969. Some demographic and genetic consequences
of environmental heterogeneity for biological control.
Bulletin of the ESA 15:237–240.
Levins, R. 1970. Extinction. Pages 77–107 in M. Gerstenhaber,
editor. Some mathematical questions in biology. American
Mathematical Society, Providence, Rhode Island, USA.
Macaya, E. C., S. Boltaña, I. A. Hinojosa, J. E. Macchiavello,
N. A. Valdivia, N. R. Vásquez, A. H. Buschmann, J. A.
Vásquez, J. M. Alonso Vega, and M. Thiel. 2005. Presence of
sporophylls in floating kelp rafts of Macrocystis spp.
(Phaeophyceae) along the Chilean Pacific coast. Journal of
Phycology 41:913–922.
Magrach, A., A. R. Larrinaga, and L. Santamarı́a. 2012.
Effects of matrix characteristics and interpatch distance on
functional connectivity in fragmented temperate rainforests.
Conservation Biology 26:238–247.
McCauley, D. E., and J. Raveill, and J. Antonovics. 1995.
Local founding events as determinants of genetic structure in
a plant metapopulation. Heredity 75:630–636.
Moilanen, A., and M. Nieminen. 2002. Simple connectivity
measures in spatial ecology. Ecology 83:1131–1145.
Newman, M. 2006. Modularity and community structure in
networks. Proceedings of the National Academy of Sciences
USA 103:8577–8582.
Pannell, J., and D. Obbard. 2003. Probing the primacy of the
patch: what makes a metapopulation? Journal of Ecology 91:
485–488.
Pastor-Satorras, R., and A. A. Vespignani. 2001. Epidemic
spreading in scale-free networks. Physical Review Letters 86:
3200–3203.
Ecology, Vol. 95, No. 2
Prugh, L. R. 2009. An evaluation of patch connectivity
measures. Ecological Applications 19:1300–1310.
Reed, D., B. Kinlan, P. Raimondi, L. Washburn, B. Gaylord,
and P. Drake. 2006. A metapopulation perspective on the
patch dynamics of giant kelp in Southern California. Pages
352–386 in J. Kritzer and P. Sale, editors. Marine
metapopulations. Academic Press, San Diego, California,
USA.
Reed, D. C., D. R. Laur, and A. W. Ebeling. 1988. Variation in
algal dispersal and recruitment: the importance of episodic
events. Ecological Monographs 58:321–335.
Reed, D. C., A. Rassweiler, and K. K. Arkema. 2008. Biomass
rather than growth rate determines variation in net primary
production by giant kelp. Ecology 89:2493–2505.
Reed, D. C., A. Rassweiler, M. H. Carr, K. C. Cavanaugh,
D. P. Malone, and D. A. Siegel. 2011. Wave disturbance
overwhelms top-down and bottom-up control of primary
production in California kelp forests. Ecology 92:2108–2116.
Scott, J. 2000. Social network analysis. A handbook. Sage
Publications, London, UK.
Snäll, T., J. Ehrlén, and H. Rydin. 2005. Colonization–
extinction dynamics of an epiphyte metapopulation in a
dynamic landscape. Ecology 86:106–115.
Theobald, D. M., J. B. Norman, and M. R. Sherburne. 2006.
FunConn v1 user’s manual: ArcGIS tools for functional
connectivity modeling. Natural Resources Ecology Lab,
Colorado State University, Fort Collins, Colorado, USA.
Thomas, C., and W. Kunin. 1999. The spatial structure of
populations. Journal of Animal Ecology 68:647–657.
Turner, M. G., R. H. Gardner, and R. V. O’Neill. 2001.
Landscape ecology in theory and practice: pattern and
process. Springer, New York, New York, USA.
Verheyen, K., M. Vellend, H. Van Calster, G. Peterken, and M.
Hermy. 2004. Metapopulation dynamics in changing landscapes: a new spatially realistic model for forest plants.
Ecology 85:3302–3312.
Watson, J. R., D. A. Siegel, B. E. Kendall, S. Mitarai, A.
Rassweiller, and S. D. Gaines. 2011. Identifying critical
regions in small-world marine metapopulations. Proceedings
of the National Academy of Sciences USA 108:E907–1.
Zalesky, A., A. Fornito, and E. T. Bullmore. 2010. Networkbased statistic: identifying differences in brain networks.
NeuroImage 53:1197–207.
SUPPLEMENTAL MATERIAL
Appendix A
A figure showing the sensitivity of the modularity approach to changing the minimum modularity threshold (Ecological Archives
E095-028-A1).
Appendix B
A figure showing the sensitivity of the modularity approach to changing the maximum linkage distance (Ecological Archives
E095-028-A2).
Appendix C
Three figures showing the actual logistic relationships between colonization/extinction and area/connectivity for the contiguity
(C1), separation-distance (C2), and modularity (C3) approaches (Ecological Archives E095-028-A3).